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Axiom ax-mulass 7509
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7469. Proofs should normally use mulass 7534 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 7409 . . . 4 class
31, 2wcel 1439 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1439 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1439 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 925 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 7416 . . . . 5 class ·
101, 4, 9co 5666 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 5666 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 5666 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 5666 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1290 . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Colors of variables: wff set class
This axiom is referenced by:  mulass  7534
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